Chapter 8, Models and theories of nuclear physics Video Solutions, Nuclear and Particle Physics: An Introduction

Problem 1

Assume that in the shell model the nucleon energy levels are ordered as shown in Figure 8.4. Write down the shell model configuration of the nucleus ${ }_{3}^{7} \mathrm{Li}$ and hence find its spin, parity, and magnetic moment

Laurent Bergeron

Numerade Educator

02:34

Problem 2

A certain odd-parity shell-model state can hold up to a maximum of 16 nucleons. What are its values of $j$ and $l ?$

Shahab Ullah

Numerade Educator

01:31

Problem 3

The ground state of the radioisotope ${ }_{9}^{17} \mathrm{~F}$ has spin-parity $j^{P}=5 / 2^{+}$and the first excited state has $j^{P}=1 / 2^{-}$. By reference to Figure $8.4$, suggest two possible configurations for the latter state.

Suzanne W.

Numerade Educator

02:40

Problem 4

What are the configurations of the ground states of the nuclei ${ }_{41}^{93} \mathrm{Nb}$ and ${ }_{16}^{33} \mathrm{~S}$ and what values are predicted in the single-particle shell model for their spins, parities, and magnetic dipole moments?

Eduard Sanchez

Numerade Educator

01:41

Problem 5

Show explicitly that a uniformly charged ellipsoid at rest with total charge $Z e$ and semi-axes defined in Figure $2.17$ has a quadrupole moment $Q=2 Z\left(a^{2}-b^{2}\right) / 5$.

Manik Pulyani

Numerade Educator

03:09

Problem 6

The ground state of the nucleus ${ }_{67}^{165}$ Ho has an electric quadrupole moment $Q \approx 3.5 \mathrm{~b}$. If this is due to the fact that the nucleus is a deformed ellipsoid, use the result of Question $8.5$ to estimate the sizes of its semi-major and semi-minor axes.

Chai Santi

Numerade Educator

01:32

Problem 7

The decay ${ }_{90}^{226} \mathrm{Th}\left(0^{+}\right) \rightarrow{ }^{222} \mathrm{Ra}\left(0^{+}\right)+\alpha$ has a $Q$-value of $6.451 \mathrm{MeV}$ and a half-life of $30.57 \mathrm{~min}$. If the frequency and probability of forming alpha particles (see (8.53)) for this decay are the same as those for the decay ${ }_{90}^{228} \mathrm{Th}\left(0^{+}\right) \rightarrow{ }_{88}^{224} \mathrm{Ra}\left(0^{+}\right)+\alpha$, estimate the half-life for the $\alpha$ decay

Zulfiqar Ali

Numerade Educator

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Problem 8

The ${ }_{92}^{238} \mathrm{U}$ is unstable and decays via alpha emission. (a) Calculate the height of the Coulomb barrier shown in Figure $8.2$ for this decay. (b) What is the distance from the centre of the nuclear potential beyond which the kinetic energy of the $\alpha$-particle is positive?

Lainey Roebuck

Numerade Educator

02:08

Problem 9

Three nuclei A, B, C are radioactive isotopes of the same element and all decay via $\alpha$-emission. The half-lives $t_{1 / 2}$ and ranges $r$ of the alpha particles for $\mathrm{A}$ and $\mathrm{B}$ are:
$$
\mathrm{A}\left(t_{1 / 2}=10^{3} \mathrm{yr}, r=3 \mathrm{~cm}\right), \quad \mathrm{B}\left(t_{1 / 2}=10^{2} \text { days, } r=4 \mathrm{~cm}\right) .
$$
If the range of the $\alpha$ particles from the decay of $\mathrm{C}$ is $r=6 \mathrm{~cm}$, estimate its half-life.

Mahendra K

Numerade Educator

01:44

Problem 10

The reaction ${ }^{34} \mathrm{~S}(p, n)^{34} \mathrm{Cl}$ has a threshold proton laboratory energy of 6. $45 \mathrm{MeV}$. Calculate nonrelativistically the upper limit of the positron energy in the $\beta$ decay of ${ }^{34} \mathrm{Cl}$, given that the mass difference between the neutron and the hydrogen atom is $0.78 \mathrm{MeV}$.

Mayukh Banik

Numerade Educator

03:57

Problem 11

To determine the mass of the electron neutrino from the beta decay of tritium requires measurements of the electron energy spectrum very close to the endpoint, where there is a paucity of events (see Figure 8.12). Obtain a rough estimate of the fraction of electrons with kinetic energies within $5 \mathrm{eV}$ of the endpoint by ignoring the Fermi screening factor and approximating the spectrum as a function of kinetic energy $T$ by $\mathrm{d} \omega / \mathrm{d} T=T^{1 / 2}\left(T_{0}-T\right)^{2}$, where $T_{0}$ is the endpoint. You may assume the integral
$$
\int_{a-\varepsilon}^{a} x^{1 / 2}(a-x)^{2} \mathrm{~d} x \approx \frac{1}{3} a^{1 / 2} \varepsilon^{3}, \quad \varepsilon \ll \mathrm{a} .
$$

Stanley Enemuo

Numerade Educator

02:04

Problem 12

Use the approximation given in Problem $8.11$ for the kinetic energy spectra of $\beta$ decays with very low-energy endpoints $T_{0}$ to show that in these cases the mean kinetic energy is $T_{0} / 3$.

Narayan Hari

Numerade Educator

02:08

Problem 13

The ground state of ${ }_{35}^{73} \mathrm{Br}$ has $J^{P}=1 / 2^{-}$and the first two excited states have $J^{P}=5 / 2^{-}(26.92 \mathrm{keV})$ and $J^{P}=3 / 2^{-}(178.1 \mathrm{keV}) .$ List the possible $\gamma$-transitions between these levels and estimate the half-life of the $3 / 2^{-}$ state.

Suzanne W.

Numerade Educator

03:42

Problem 14

The hadrons $\Sigma^{0}$ and $\Delta^{0}$ can both decay via photon emission: $\Sigma^{0}(1193) \rightarrow \Lambda(1116)+\gamma \quad$ (branching ratio $\left.\sim 100 \%\right) ; \quad \Delta^{0}(1232) \rightarrow n+\gamma$ (branching ratio $0.56 \%$ ). If the lifetime of the $\Delta^{0}$ is $0.6 \times 10^{-23} \mathrm{~s}$, estimate the lifetime of the $\Sigma^{0}$. The $J^{P}$ values of the hadrons are given in Figure $3.16$.

Narayan Hari

Numerade Educator

01:39

Problem 15

Use the Weisskopf formulas (8.88a) and (8.88b) to calculate the radiative width $\Gamma_{\gamma}(\mathrm{E} 3)$ expressed in a form analogous to (8.89).

Mahnoor Amin

Numerade Educator